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CHAPTER 2
LITERATURE REVIEW
Design optimization is the application of numerical algorithms and
techniques to engineering systems to assist designers in improving the
system's performance. Optimization is a process of maximizing or minimizing
a desired objective function while satisfying the prevailing constraints.
Typical engineering systems are described by a very large number of
variables, and it is the designer's task to specify appropriate values for these
variables. Optimization techniques can be applied during the product
development stage to ensure that the finished design will have high
performance, high reliability, low weight, and/or low cost. Alternatively,
optimization methods can be applied to existing products to achieve potential
design improvements. Design optimization techniques have been used in a
number of fields, including automobile design, naval architecture, electronics,
computers, and electricity distribution. However, the largest number of
applications has been in the field of aerospace engineering, such as aircraft
and spacecraft design.
An overview of design optimization considering uncertainty and an
insight into the Reliability Based Design Optimization problem is presented in
this chapter. First, a review of traditional design optimisation is presented and
several mathematical models of uncertainty in engineering are introduced.
Then, a literature survey of the existing formulations of non-deterministic
design optimization problems is reported. The concepts of design
optimization under uncertainty with emphasis on reliability based design, as
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well as the fundamental differences between the robust design and the
Reliability-based Design Optimization (RBDO) are presented. A review of
“RBDO formulation, methodologies and applications” is then presented. The
application design optimization for composite laminates is also reported.
2.1 TRADITIONAL DESIGN OPTIMIZATION
Design Optimization is a process of obtaining the best parameters
such as thickness, height, length, module, number of teeth etc. of a component
or a system under certain given circumstances (Belegundu 1981). In a
traditional design optimization problem, the free parameters that need to be
determined to obtain the desired performance are known as the design
parameters. The function for evaluating the merits of a design is called the
objective function. Generally, a number of restrictions must be satisfied in a
design optimization problem. These restrictions define the feasible domain in
the design variable space and are referred to as the design constraints.
Additionally, bound limits may be imposed on the design variables and they
are known as side constraints. In a design optimization problem, the objective
function and the constraints are often expressed as implicit functions of the
design variables and the evaluation of these functions generally involves
numerical simulation techniques such as the finite element method
(Rao 1996).
The traditional deterministic design optimization problem is
mathematically represented as given below (Belegundu and
Chandrupatla 2003).
find d
minimizing f(d)
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subject to gi(d) ≤ 0 (i = 1, 2, …., k),
hj(d) = 0 (j = 1, 2, …., p)
dL < d < dU
where d Є Rn+1 is the vector of design variables, f the objective
function, gi (i = 1, 2, ..., k) the inequality constraint functions, hj (j=1, 2, .., p),
the equality constraints and dL and dU denote the lower and upper bound
limits of the design variables, respectively. The design variables can be the
parameters defining the geometrical dimensions, the shape or the topology of
the structure. In practical applications, it is common to make use of the design
variable linking technique to reduce the number of the independent design
variables by imposing a relationship between coupled design parameters. The
objective function and the constraint functions are usually the cost, the
material volume/weight, performances such as nodal displacements, stresses,
natural frequencies, and buckling loads. Since these functions are typically
implicit functions of the design variables, a structural analysis such as the
finite element analysis must be performed whenever their values are required.
In the deterministic formulation of the design optimization problems, the
design variables and other parameters are assumed to be deterministic and the
objective functions as well as the constraints are determined based on their
nominal values (Arora 1990). A solution to an optimization problem specifies
the values of the decision variables, and also the value of the objective
function which must be feasible and optimal. A feasible solution satisfies all
constraints and an optimal solution is the one that minimizes the objective
function. Figure 2.1 shows the feasible region which satisfies all the
constraints (Deb 2003).
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Figure 2.1 Feasible region
2.1.1 Classification
Design Optimization problems are classified based on the existence
of constraints, the number of variables, nature of variables, nature of
expressions, and permitted values of the design variables. They are
summarized as below:
According to the existence of constraints, an optimization problem
can be classified as a constrained or unconstrained problem (Johnson 1961).
Based on the nature of the design variables encountered,
optimization problems can be classified into Static and Dynamic Optimization
problems (Pappalambras and Wilde 1988).
Another important classification of optimization problems is based
on the nature of expressions for the objective function and the constraints.
According to this, optimization problems can be classified as linear,
nonlinear, geometric and quadratic programming problems (Arora 1990).
g (X) ≤ 0
h (X) ≥ 0
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Depending on the values permitted for the design variables,
optimization problem may be classified as integer and real-valued
programming problems (Rao 1996).
According to the types of the design parameters to be considered,
design optimization problems can be broadly classified into three categories:
Sizing optimization: design variables are geometrical
dimensions such as cross sectional areas of truss members,
beam section parameters and plate thickness (Arora 1990,
Haftka and Gurdal 1992, Deb 2003).
Shape optimization: design variables are the geometry
parameters describing the shape of the designed parts
(Belegundu and Rajan 1988, Gu and Cheng 1990,
Broudiscou et al 1995).
Topology and layout optimization: the number and locations
of voids in a continuous structure or the number and
connectivity of members in a discrete structure (e.g. truss
and frame structure) are to be determined (Kirsch 1989,
Bendsoe and Motasoarse 1993, Buhl et al 2000).
Research on the methods and applications of design optimization
has increased rapidly during the past decades. A variety of numerical
techniques have also been developed and applied to both linear and nonlinear
problems (Arora 1990, Bhul et al 2000).
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2.1.2 Methodologies
Basically, the solution methods for design optimization problems
can be classified into Optimality Criteria (OC) methods and Mathematical
Programming methods. In the Optimality Criteria method (Zou and
Haftka 1995), the optimality conditions for a given type of problem are
derived based on the Karush-Kuhn-Tucker condition or by heuristic
assumptions, and then the optimal design satisfying these condition is to be
sought using different forms of resizing rules. Such methods are recognized to
be especially efficient for problems involving a large number of design
variables.
The Mathematical Programming method may be broadly classified
into gradient-based methods (requiring derivatives of the functions) (Johnson
1961, Arora 1990) and non-gradient or direct methods (requiring no
derivatives) (Fox 1971, Rao 1996, Deb 2003).
The use of the gradient-based method for minimization is first
presented by Cauchy. Modern optimization methods are pioneered by
Courant’s paper on penalty functions, Dantzig’s paper on the simplex method
for linear programming and Karush, Khun and Tucker who derived the
“KKT” optimality conditions for constrained problems (Johnson 1961).
Particularly in the 1960s, several numerical methods to solve nonlinear
optimization problems were developed. Mixed integer programming received
an impetus from the branch and bound technique, originally developed by
Land and Doig, and the cutting plane method by Gomory (Fox 1971).
Methods of unconstrained minimization include the Conjugate gradient
methods of Fletcher and Reeves, and the variable metric methods of
Davidon-Fletcher-Powell (DFP) (Siddall 1972).
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The Constrained optimization method is pioneered by Rosen’s
gradient Projection method, Zoutendijik’s method of feasible directions, the
generalized reduced gradient method by Abadie, Carpentier and Hensgen and
Fiacco and McCormick’s SUMT techniques (Siddall 1972). The traditional
interval search methods, using Fibonacci numbers or the golden section ratio
are followed by the efficient hybrid polynomial-interval methods of Brent.
Sequential Quadratic Programming (SQP) methods for constrained
minimization are then developed. The Development of interior methods for
linear programming started with the work of Karmarkar in 1984 (Papalambras
and Wilde 1988).
In the 1960s, side-by-side with developments in gradient-based
methods, there were developments in non-gradient methods, principally
Rosenbrock’s method of orthogonal directions, the pattern search method of
Hooke and Jeeves, Powell’s method of Conjugate directions, the simplex
method of Nedler and Meade and the method of Box (Haug and Arora 1979).
Most recent among the direct methods are genetic algorithms (Holland 1975,
Goldberg 2002) and the simulated annealing algorithm which originated from
Metropolis. Special methods that exploit some particular structure of a
problem have also been developed (Rao 1996). Dynamic programming
originated from the work of Bellman, who stated the principle of optimal
policy for system optimization. Geometric programming originated from the
work of Duffin Peterson, Zener. Pareto optimality was developed in the
context of multiobjective optimization.
In addition to these conventional methods, some innovative
approaches using analogies of physics and biology, such as Simulated
Annealing, Genetic Algorithm and Evolutionary Algorithms
(Papadrakakis et al 1998, Deb 2001), are also employed for the solution of
global optimization problems. These approaches are characterized by
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gradient-free methods and utilize only function values. Generally, these
algorithms require a large number of function evaluations to achieve
convergence, and thus have limited use in applications involving complicated
designs.
Different methods of design optimization are widely used in the
design of engineering structures for the purpose of improving the performance
and reducing their costs. The use of design optimization techniques has
rapidly increased, mainly due to the development of sophisticated computing
techniques and the extensive applications of the finite element method
(Deb 2003). Moreover, recently, it is widely recognized that design
optimization methodologies should account for the stochastic nature of
engineering systems, and that concepts and methods of life-cycle engineering
should be used to obtain a cost-effective design during a specified time
horizon. To ensure high reliability and safety, uncertainties inherent to or
encountered by the product during the entire life cycle must be considered and
treated in the design process. The various types of uncertainty, the
mathematical models of uncertainty reported in literature, and the
optimization methodologies which include these uncertainties are described
below.
2.2 TYPES OF UNCERTAINTY
Uncertainty is an acknowledged phenomenon in the natural and
technological worlds. Engineers are continually faced with uncertainties in
their designs. However, there is no unique definition of uncertainty. A useful
functional definition of uncertainty is: the information/knowledge gap
between what is known and what needs to be known for optimal decisions,
with minimal risk. Uncertainties can be modeled or quantified using the
probability theory, convex models of uncertainty and fuzzy set theory
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(Dubois and Prade 1988, Fedrizzi et al 1994), Dempster-shafer theory
(Liu et al 2006).
The design uncertainties include variations in certain parameters
which are either controllable (e.g. dimension) or uncontrollable (e.g. material
properties), and model uncertainties and error associated with the simulation
based design. In general, a distinction can be made between aleatory
uncertainty (also referred to as stochastic uncertainty, irreducible uncertainty,
inherent uncertainty, and variability), epistemic uncertainty (also referred to
as reducible uncertainty, subjective uncertainty, model form uncertainty or
simply uncertainty), and numerical uncertainty (also known as error)
(Zissimos and Zhou 2006). Oberkampf et al (2004) have described various
methods for estimating the total uncertainty by identifying all possible sources
of variability, uncertainty, and error in mathematical models and simulation
tools.
2.2.1 Aleatory uncertainty
Aleatory uncertainty (originating from the Latin aleator or
aleatorius, meaning dice thrower) is used to describe the inherent spatial and
temporal variation associated with the physical system or the environment
under consideration, as well as the uncertainty associated with the measuring
device (Ben-Haim and Elishakoff 1990). Sources of aleatory uncertainty can
be represented as randomly distributed quantities. Aleatory uncertainty is
also referred to in the literature as variability, irreducible uncertainty,
inherent uncertainty, and stochastic uncertainty. Aleatory uncertainty can
occur in the form of manufacturing tolerances (Elishakoff et al 1994).
Probability distributions can be used to model such uncertainty.
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2.2.2 Epistemic uncertainty
Epistemic uncertainty (originating from the Greek episteme,
meaning knowledge) is defined as any lack of knowledge or information in
any phase or activity of the modeling process. Examples of sources of
epistemic uncertainty are scarcity or unavailability of experimental data for
fixed (but unknown) physical parameters, limited understanding of complex
physical processes or interactions of processes in the engineering system, and
the occurrence of fault sequences or environment conditions not identified for
inclusion in the analysis of the system (Zissimos and Zhou 2006). Epistemic
uncertainty can be either parametric or model-based. Parametric uncertainty
is associated with the uncertain parameter for which the information available
is sparse or inadequate. A model form of the uncertainty, also known as tool
uncertainty, is associated with improper models of the system due to lack of
knowledge of the physics of the system (Mahadevan and Ramesh 2006).
Fuzzy sets are used to model such uncertainty.
2.2.3 Numerical Uncertainty
Error (numerical uncertainty) is commonly associated with the
numerical models used for simulations and modeling. In the convergence of a
coupled system analysis, round-off errors, truncation errors and errors
associated with the solution of Ordinary Differential Equation (ODE) and
Partial Differential Equations (PDE) (Huibin et al 2006) are considered as
numerical uncertainities.
2.3 MATHEMATICAL MODELS OF UNCERTAINTY
The formulation of a design optimization problem under
uncertainty is closely related to the modeling of the uncertainty. There exist
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various mathematical models of uncertainty, when dealing with design
optimization problems. The existing models can be classified into
probabilistic models e.g. stochastic randomness, and non-probabilistic models
including interval set, convex modeling, fuzzy set and leveled noise factors. A
brief introduction to these uncertainty models is given below.
2.3.1 Randomness
The prevailing model for uncertainties in engineering design is
stochastic randomness (Doltsinis 1999, Schueller 2001). The Probability
Density Function (PDF) and Cumulative Distribution Function (CDF) are
used to define the occurrence properties of uncertain quantities which are
random in nature. Randomness accounts for most of the uncertainties in
engineering problems (Zou et al 2002). In computational engineering
problems, the model errors and the uncertainties that arise from incomplete
knowledge about the system, are often regarded as random uncertainties as
well. In practical design problems the randomness of the uncertain parameters
is often modeled as a set of discretized random variables (Youn and
Choi 2004). The statistical description of a random variable X can be
completely described by a cumulative density function F(x) or a probability
density function (PDF) f(x) defined as
dx(x)fx)P(X(x)F X
x
X (2.1)
where P(.) is the probability that an event will occur. The probability
distribution of the random variable X can be also be characterized by its
statistical moments (Haldar and Mahadevan 2000). The most important
statistical moments are the first and second moment known as mean value
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μ(X), also referred to as expected value and denoted by E(X), and variance
denoted by Var(X) or σ2(X), respectively, as given by
dx)x(xf)x(xdF)X(E)X( XX
, (2.2)
and
dxxfXxxdFXxX XX )())(()())(()( 222
(2.3)
Probability distributions such as Normal, lognormal and Weibull
are the most commonly used distributions in design optimization problems.
Nevertheless, precise information on the probabilistic distribution of the
uncertainties is sometimes scarce or even absent. Moreover, some
uncertainties are not random in nature and cannot be defined in a probability
framework (Nikolaidis et al 2008). For these reasons, non-probabilistic
methods for modeling of uncertainties have been developed in recent years.
These methods do not require a priori assumptions on PDFs for the
description of uncertain variables.
2.3.2 Interval Set
An Interval set is used to model uncertain but non-random
parameters. These uncertainties are assumed to be bounded within a specified
interval and a small variation of the interval parameter is treated as a
perturbation around the midpoint of this interval, allowing the interval
perturbation method to be used for the analysis of the performance variation
(Rao and Berke 1997, Qui and Elishakoff 1998). Using the so-called
anti-optimization techniques, the least favourable response can be determined
under the assumption of small variations. The term anti-optimization is
referred to as the task of finding the worst-scenario of a given problem
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(Elishakoff et al 1994). The methods based on the interval set do not allow for
the distinction of the more or less probable occurrence of the variables
(Lombardi and Haftka 1998). Moreover, it is difficult to consistently define
bounded intervals for the uncertainties, without a confidence level.
2.3.3 Convex Modeling
To overcome the difficulties when data are insufficient to conduct a
reliability analysis using conventional probabilistic approaches, the
worst-case scenario analysis based on Convex Modeling can be formulated
(Ben-Haim and Elishakoff 1990). Convex Modeling is connected to
uncertain-but-bounded quantities. In this method, the uncertainty which has
bounded values is assumed to fall into a multi-dimensional ellipsoid or
hypercube. In some sense, the convex model can be regarded as a natural
extension of the interval set model. By virtue of the Convex Model theory,
the worst-case performance of the design is determined using the
anti-optimization technique (Yoshikawa et al 1998). This method has been
proved to be more advantageous than the traditional worst-case approach,
where all the possible combinations of the extreme values of the uncertain
parameters need to be examined, so that the worst-case scenario can be
determined (Pantelides and Ganzerli (1998).
2.3.4 Fuzzy Set
The fuzzy set theory has been developed as a mathematical tool for
quantitative modeling of the uncertainty associated with vagueness in
describing subjective judgments under uncertainty using linguistic
information (Rao 1987). In the fuzzy set method for engineering design
problems, the uncertainty is modeled as fuzzy numbers rather than random
values with certain distribution (Sakawa 1993). In other words, the fuzzy set
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theory presents a possibility rather than a probability description of the
uncertainty. The fuzzy analysis method has been used to deal with certain
problems such as design analysis under uncertain loading conditions (Gerhard
and Haftka 1998). Venter and Haftka (1999) used response surface
approximation in fuzzy set based optimization. Liu et al (2006) proposed
possibility-based design optimization methods for design problems with fuzzy
data.
2.3.5 Leveled Noise Factors
In Taguchi’s robust design methodology (Tsui 1992), the system
uncertainties are modeled as leveled noise factors. Here, no a priori
assumptions on the statistics of the uncertainties are required. Lee et al (1996)
devised a robust design for unconstrained optimization problems using the
Taguchi method. Following the method of experimental design, the system
outputs are examined at planned combinations of the discrete levels of these
noise factors. Thus, the interactions between the system performance and
noise factors can be explored (Montgomery 2001). Wang and Kodiyalam
(2002) proposed an efficient method for a probabilistic and robust design with
non-normal distributions.
The various design optimization methods that incorporate
uncertainty are given in the following Section.
2.4 DESIGN OPTIMIZATION UNDER UNCERTAINTY
Conventional design procedures accounting for system
uncertainties are based on safety factors. This method often produces far too
conservative designs. Use of deterministic methods to reduce cost or weight
may result in systems that are vulnerable to variability and uncertainty,
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because these methods operate on very tight margins. Moreover, in the design
of novel structures or products, little prior knowledge for determining an
appropriate safety factor is available. More sophisticated formulations
incorporating system uncertainty into design optimization have been reported
in the past on the basis of various mathematical models of uncertainty (Tonon
and Bernardini 1998).
2.4.1 Worst Case Scenario-based Design Optimization
In some non-deterministic design optimization problems, the design
against system failure is based on the worst case analysis. In practical
applications of this approach, the convex model or interval set can be used to
model the system uncertainties. Elishakoff et al (1994) applied this method to
the optimal design problem considering bounded uncertainty.
Yoshikawa et al (1998) presented a formulation to evaluate the worst case
scenario for a homology design, caused by uncertain fluctuation of loading
conditions using the convex model of uncertainty. In these approaches, the
anti-optimization strategy is adopted to determine the least favorable
combination of the parameter variations and the problem is then converted
into a deterministic Min-max optimization (Qiu and Elishakoff 1998). Here,
no probability density function of the input variables is required. The validity
of the proposed method is demonstrated by its application to the design of
simple truss structures (McWilliam 2000).
Lombardi and Haftka (1998) combined the worst-case scenario
technique of anti-optimization and the optimization techniques in the design
that considers uncertainty. The proposed method is suitable in particular for
uncertain loading conditions. Since a complete optimization routine needs to
be nested for the worst case analysis at each design optimization cycle, this
approach may become prohibitively expensive when many uncertain
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parameters are presented in the problem. Additionally, this design technique
often results in too conservative designs.
2.4.2 Robust Design Optimization
In robust design, the performance of the system is required to be
less sensitive to the random variations induced at different stages of the
product’s life cycle. Robust design is an engineering methodology for the
optimal design of products and process conditions that are less sensitive to
system variations (Ranganathan 1990). It has been recognized as an effective
design method to improve the quality of the product/process. Among the three
stages of engineering design, viz. conceptual, parameter and tolerance design,
robust design may be involved in the stages of parameter design and tolerance
design.
For design optimization problems, the performance function
defined by design objectives or constraints may be subject to large scatter at
different stages of the service life-cycle. It can be expected that this may be
more crucial for systems with nonlinearities (Rao 1992). Such scatters may
not only significantly worsen the product quality and cause deviations from
the desired performance, but may also add to the product’s life-cycle costs,
including inspection, repair and other maintenance costs (Lee et al 1996).
From an engineering perspective, well-designed structures minimize these
costs by performing consistently in the presence of uncontrollable variations
during the whole life-cycle. In other words, excessive variations of the system
performance indicate the poor quality of a product. This raises the need for
robust design (Montgomery 2001).
One possible way to decrease the scatter of the system
performance, is to reduce or even eliminate the scatter of the input
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parameters, which may either be practically impossible or add much to the
total costs of the structure; another way is to find a design in which the system
performance is less sensitive to the variation of parameters without
eliminating the cause of parameter variations, as in robust design (Rao and
Berk 1997).
The robust design optimization approach not only shifts the
performance mean to the target value, but also reduces the product’s
performance variability, achieving Six-sigma level robustness on the key
product performance characteristics with respect to the quantified variation.
The Taguchi methodology (Montgomery 2001) indicates, that by conducting
planned experiments under some assumptions, uncontrollable or noise
variables can be precisely controlled, and thereby the designer can choose the
levels of controllable variables to accomplish a robust system that is
insensitive to inevitable changes of the noise variables (Huibin et al 2006).
2.4.3 Fuzzy Set based Design Optimization
The fuzzy set theory has been initially used by Rao (1987) to
handle design optimization under uncertainties. The design problem involves
maximizing the safety level of a structure. The response surface methodology
is also used throughout the design process to reduce the computational effort
(Ranganathan 1990, Rao 1992). A random set approach has been proposed
by Tonon and Bernardini (1998) as an extension of the fuzzy set method for
the design optimization problem which is characterized by imprecise or
incomplete observations on the uncertain design parameters. Gerhard and
Haftka (1998) used the fuzzy set theory for modeling the uncertainty
associated with the design with future materials in the aircraft industry.
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In the fuzzy set-based design optimization, the vague quantities
which cannot be clearly defined in a system, are characterized by membership
functions (Venter and Haftka 1999). In this context the possibility of system
failure is restricted to the optimal design. Since this method is featured as a
non-probabilistic description of system reliability, it can be regarded as a
possibility-based approach. In a similar way, as in Reliability Based Design
Optimization, this approach focuses exclusively on the issue of the system
safety with the purpose of avoiding system catastrophe in the presence of
parameter uncertainties.
2.4.4 Reliability Based Design Optimization
The term reliability-based design optimization is used, in a narrow
sense, exclusively to the optimal design where the cost function of the
problem is to be minimized under the observance of probabilistic constraints
instead of conventional deterministic constraints (Rackwitz et al 1995). Until
recently, the RBDO has been the only way of taking account of uncertainty in
design optimization problems. When the occurrence of the catastrophic failure
of the system or component is crucial, the design optimization problem is
usually characterized as a problem of reliability-based design optimization. In
this framework, the probability of failure is involved in the constraint
conditions of the design optimization problems. The failure of a system or a
component is defined with limit state functions (Kuschel and Rackwitz 2000).
From the theoretical point of view, reliability-based design optimization has
been a well-established concept. Prior to the reliability analysis, the statistical
characteristics of the random quantities are first defined by suitable
probability distributions.
In RBDO the probability of failure is evaluated by numerical
procedures such as the Monte Carlo Simulation, the First Order Reliability
36
Method and Second Order Reliability Method (Rackwitz 2001). In the direct
Monte Carlo simulation or Importance Sampling method, the probability of
failure is derived from the test data of a large number of samples
(Mohsine et al 2006). In the First Order Reliability Method (FORM), and the
Second Order Reliability Method (SORM) or the Advanced Mean Value
method, an additional nonlinear constrained optimization procedure is
required for locating the design point or Most Probable Point of failure (MPP)
and thus the reliability based design optimization becomes a two-level
optimization process with lengthy calculations of sensitivity analysis in the
inner loop for locating the MPP (Nikolaidis et al 2008).
2.4.5 Differences between robust design and RBDO
Compared to the RBDO, robust design is a relatively new issue in
engineering design. As representative non-deterministic design optimization
formulations, both of them aim at incorporating random performance
variations into the optimal design process, and therefore they are sometimes
not clearly distinguished in the literature. However, the two approaches differ
in some fundamental aspects, despite the fact that the optimal solution of the
robust design often exhibits an increased reliability. First of all, the robustness
is assessed by the measure of the performance variability around the mean
value, most often by its standard deviation, whereas reliability is connected to
the probability of failure occurrence (Figure 2.2).
In general, RBDO is concerned more with satisfying the reliability
requirements under known probabilistic distributions of the input, and less
concerned with minimizing the variation of the performance function, while
the robust design aims to reduce the system variability to unexpected
variations (Kaymaz and McMahon 2004). In RBDO, the objective function is
to be minimized under the observance of probabilistic constraints. However,
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in robust design optimization, the objective function usually involves the
performance variations, and the design constraints may be simply defined by
the variance (Anukal and Mahadevan 2005). Actually, RBDO is usually
accomplished by moving the mean of the performance as depicted in
Figure 2.3, whereas robust design is often implemented by diminishing the
performance variability, as shown in Figure 2.4.
Figure 2.2 Difference between robustness and reliability
Figure 2.3 RBDO strategy
Prob
abili
ty D
ensit
y Fu
nctio
n P(
f)
Performance f
Probability of failure
Limit State function
μf
σf
Prob
abili
ty D
ensi
ty
Func
tion
P(f)
Performance f
μf
Less reliable More reliable
38
Figure 2.4 Robust design strategy
Secondly, in RBDO particular care is paid to the issue of system
safety in extreme events, while in robust design more emphasis is put on the
behavior under everyday fluctuations of the system during the whole service
life (Zang et al 2002).
2.5 RBDO FORMULATION, METHODOLOGY AND
APPLICATIONS – A REVIEW
Reliability Based Design Optimization (RBDO) methodologies not
only provide improved design but also a confidence range of the simulation
based optimum design (Huibin et al 2006, Nicholaidis et al 2008). The basic
idea in reliability based design optimization, is to employ numerical
optimization algorithms to obtain optimal designs ensuring reliability. There
are two Approaches of RBDO, namely: the Reliability Index Approach (RIA)
and the Performance Measure Approach (PMA) (McDonald and
Mahadevan 2008).
Prob
abili
ty D
ensi
ty
Func
tion
P(f)
Performance f
μf
σf Less Robust More Robust
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2.5.1 Reliability Index Approach
The Reliability Index Approach (RIA) was first introduced by
defining a probabilistic constraint as reliability (Enevoldsen and Sorensen
1994, Frangopol and Moses 1994 Yu et al 1997, Tu and Choi 1999, Alan et al
2007). Many researchers (Enevoldsen and Sorensen 1994, Chandu and Grandi
1995, Yu et al 1997,Wu and Wang 1998, Grandhi and Wang 1998, Choi et al
2004) have used the reliability index evaluated in the traditional reliability
analysis to prescribe the probabilistic constraint. For a reliability based
design, a performance function can be defined as G = R-S where R and S are
statistically independent and normally distributed random variables of the
resistance and load measurements of the structure. Typically, R can be the
yield stress and S the maximum Von Mises stress. The G function is also
called the limit state function or failure function as shown in Figure 2.5. The
curve G = 0 divides the design space into two regions, the safe region when
G > 0 and the unsafe region when G < 0. Since R and S have a variation, G
will also exhibit variation.
Figure 2.5 Reliability Index
β
G(u)
SORM
FORM
u*=MPP
u1
u2
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The ratio (β) of the mean value of the G function (μG) and the
standard deviation of the G function (σG) is defined as safety index or
reliability index. If Φ is the cumulative distribution function and G has a
normal distribution then:
β = -Φ(1-Reliability) = μG / σG (2.4)
Pf = Φ (-β) (2.5)
where β is the distance from the origin to the Most Probable Point of Failure
(MPP). In both the First Order Reliability Method (FORM) and the Second
Order Reliability Method (SORM), the original random variables, which are
generally non-normal and correlated, are first transformed into an equivalent
set of statistically independent normal variates. A general transformation for
this purpose is the Rosenblatt transformation (Tvedt 1990, Nikolaidis 2008).
During optimization, the corresponding MPP in X-space needs to be
calculated to evaluate the probabilistic performance functions. The MPP of
failure in X-space is found by mapping u*β=ρ to the original space. If the
random variables in X-space are independent and normally distributed, then
the MPP in the original space is given by x*=μx-u*β=ρσx. If the variables have
a non-normal distribution, then the equivalent means and equivalent standard
deviations of an approximate normal distribution are computed and used in
the above expression to estimate the MPP in X-space (Rao 1992).
The RBDO problem can be formulated to maximize the Reliability
Index β while simultaneously minimizing the weight than the target value
(Type-I) as given below.
41
Maximize β
Subject to: Weight < Target-Weight
DVmin< DV < DVmax
where DV = Design Variable
The RBDO problem can also be formulated to minimize the mass
while simultaneously maximizing the Reliability Index β than the target
(Type-II) as given below:
Minimize Weight
Subject to: β > Target β
DVmin< DV < DVmax
A typical target value β commonly used in the literature is 3, which
corresponds to a failure probability of 0.00135. It is observed from the
RBDO literature that the Reliability Index Approach exhibits very slow
convergence or even divergence for some problems (Tu and Choi 1999, Tu et
al 1999, Youn et al 2003, Choi and Youn 2002, Youn and Choi 2004).
2.5.2 Performance Measure Approach (PMA)
To overcome the disadvantages of the Reliability Index Approach,
the Performance Measure Approach (PMA) is introduced by solving an
inverse problem (Palle and Michael 1982, Madsen et al 1987, Tu and Choi
1999, Youn et al 2003). A significant effort has been devoted to formulate
reliability based design optimization using the Performance Measure
Approach (Tu et al 1999).
42
The RBDO formulation for a Target Reliability of 95% using PMA
is given below.
Minimize Weight
Subject to:
P (Load > 1kN) ≤ Pfailure = 5%
P(Displacement >30 mm) ≤ Pfailure = 5%
Design Variable: Thickness ‘t’
maxmin
maxmin
iii
iii
ttt
ttt
The fact that the PMA is inherently robust and more effective is not
surprising, since it is easier to minimize a complicated cost subject to a simple
constraint expressed as the known distance (i.e., reliability index), than to
minimize a simple cost subject to a complicated constraint (Choi and
Youn 2002,Youn et al 2003). In addition, the RIA introduces an undesirable
nonlinearity, whereas the PMA does not, which becomes more serious for
non-normal distributions (Choi and Youn 2002).
The following Section explains the possible formulations of RBDO.
2.6 RBDO FORMULATIONS
There are two important concepts in RBDO formulation. They are
efficiency and robustness. An efficient formulation is one in which the
solution can be obtained faster as compared to the other formulations.
Robustness, on the other hand, means that the RBDO formulation does not
depend on the starting point. In the last two decades, researchers have
43
proposed a variety of frameworks for efficiently performing reliability based
design optimization (Shan and Wang 2008). A careful survey of the literature
reveals that the various RBDO methods can be divided into three broad
categories.
2.6.1 Double Loop Methods
A straightforward approach to solve an RBDO problem is to
conduct a double-loop optimization process in which the outer loop iteratively
selects feasible designs that approach the minimum objective, while the inner
loop evaluates the reliability constraints for each selected design
(Tu et al 1999, Youn and Choi 2004, Yang et al 2005, Jun and Mourelatos
2008). However, for complicated G-functions and objective functions, the
repeated inner-loop reliability analysis can cause the Reliability Based Design
Optimization to be prohibitively time-consuming (Nikolaidis 2008).
2.6.2 Decoupled Method or Sequential Method
Chen et al (1997) proposed a sequential RBDO methodology for
normally distributed random variables. Wang and Kodiyalam (2002)
generalized this methodology for non-normal random variables and reported
enormous computational savings when compared to the nested RBDO
formulation. Chen and Du (2002) proposed a Sequential Optimization and
Reliability Assessment methodology (SORA).
Agarwal et al (2003) extended this methodology for
multidisciplinary systems. The basic concept behind the sequential RBDO
technique is to decouple the upper level optimization from the reliability
analysis to avoid a nested optimization problem (Du and Chen 2004,
Yang et al 2005). Zou and Mahadevan (2006) proposed a direct decoupling
44
approach for efficient reliability-based design optimization. In SORA, the
boundaries of the violated constraints (with low reliability) are shifted
towards the feasible direction, based on the reliability information obtained in
the previous iteration. Therefore, a consistent reliable design is almost
guaranteed to be obtained from this framework. However, a true local
optimum cannot be guaranteed. This is because the MPP of failure for the
hard constraints is obtained at the previous design point. A shift factor, Si,
from the mean values of the random variables is calculated and is used to
update the MPP of failure for probabilistic constraint evaluation during the
deterministic optimization phase in the next iteration, as this technique varies
the mean values of the random variables. This MPP update may be inaccurate
because of the fact that as the optimizer varies the design variables, the MPP
of failure (and hence the shift factor) also changes and this is not addressed in
SORA. This may lead to spurious optimal designs (Yuan et al 2007 Jun and
Mourelatos 2008).
2.6.3 Single Loop RBDO or Unilevel RBDO
As outlined before, RBDO is typically a nested optimization
problem, requiring a large number of system analysis evaluations. The major
concern in evaluating reliability constraints is the fact that the reliability
analysis methods are formulated as optimization problems. To overcome the
difficulty of SORA, a unilevel RBDO formulation was developed by Kuschel
and Rackwitz (2000). In this method, the lower level optimization is replaced
by the corresponding first order Karush-Kuhn-Tucker (KKT) optimality
conditions of the first order reliability problem. As mentioned earlier, the
direct FORM problem can be ill conditioned, and the same may be true for
the unilevel formulation given by Rackwitz (2001). The probabilistic hard
constraints may have a zero failure probability at a particular design setting,
and hence the solution may not converge due to the hard constraints (which
45
are posed as equality constraints) not being satisfied. Moreover, the condition
under which such a replacement is equivalent to the original bi-level
formulation is not detailed in Kuschel and Rackwitz (2001). During the last
few years, researchers in the area of multidisciplinary optimization have
continuously faced the challenge to develop more efficient techniques to solve
the RBDO problem (Yang et al 2005, Harish et al 2007). A new unilevel
method is being developed which enforces the constraint qualification of the
KKT conditions and avoids the singularities associated with zero probability
of failure (Alan et al 2007, Jinghong et al 2007, Yuan et al 2007, McDonald
and Mahadevan 2008).
2.7 RBDO- RELIABILITY ESTIMATION METHODS
The methodologies used to determine the failure probability in
RBDO problems for both the approaches may be classified into two
categories, viz. Analytical methods and Simulation-based methods.
2.7.1 Analytical Reliability Methods
In this method, the accurate estimation of failure probability
requires multidimensional integration, which is difficult and highly time
consuming. Hence, approximation of the failure surface is required. The First
Order Reliability Method (FORM) (Tu and Choi 1999, Anukal and
Mahadevan 2005) uses a linear approximation of the limit-state function at
the most probable point (MPP). Consequently, for the nonlinear limit state
equation, FORM will overestimate the probability of failure, since it considers
the contribution of the region between the real limit state and the
approximation in calculating the failure probability integral (Zhao and Ono
1999). The linear approximation errors may be too large (Mitteau 1996) and
so the “Second Order Reliability Method” (SORM) is introduced, which uses
46
the first order reliability index with a correction term (Rao 1996). Studies on
the applicable ranges, history, and potential improvements and extensions of
both FORM and SORM can be found in Rackwitz (2001). Both methods
require the calculation of the MPP, which is an auxiliary optimization process
or “loop” that adds to the computational cost. Several methods have been
proposed to improve the efficiency of the “double loop” solution, namely, the
design and MPP.
Several Second-Order Reliability Methods (SORM) have also been
developed, which use parabolic approximation (Koyluolu and Neilsen 1988,
Cai and Elishakoff 1994). The Advanced Mean Value method (AMV), the
Conjucate Mean Value method (CMV) and Hybrid Mean Value method
(HMV) are the advancements of the analytical methods used in the
probabilistic constraint assessment during the RBDO process (Cruse et al
1988, Wu et al 1990, Youn et al 2003, Youn and Choi 2004). In general, the
AMV method exhibits divergence or a slow rate of convergence in addressing
a concave performance function, although it is good for a convex performance
function. Therefore, a robust and efficient hybrid mean value (HMV) method
has been proposed for the numerical solution of the inverse PMA problem
(Youn et al 2003).
Even with the HMV method, the RBDO process may not be
efficient enough to affordably obtain a reliability based optimum design for
large-scale applications or applications where design sensitivity is
unavailable. A new RBDO methodology is developed to integrate the
proposed HMV method with many MPP search algorithms such as the
Modified HL-RF and AMVFO and general optimization algorithms such as
Sequential Linear Programming (SLP) (Kuei et al 2007, Yuan et al 2007).
Sequential Quadratic Programming (SQP) and the augmented Lagrangian
47
method can be used to find the MPP (Yu et al 1997, Grandhi and Wang 1998,
Huibin et al 2006, Kuei-Yuan et al 2007).
2.7.2 Simulation-Based Reliability Methods
A variety of approximation schemes are employed to compute these
probabilities, including sampling techniques based on the Monte Carlo
Simulation (MCS) procedure. The Accuracy of MCS estimations increases
with increased sampling size, but setting low failure probability levels Pf and
dealing with costly constraint functions makes this impractical. Many
sampling techniques have been proposed to maintain the advantage of MCS
with smaller samples (Kim and Diwekar 2002).
Monte Carlo simulation methods do not require any
transformations of the random variables to an uncorrelated standard normal
space, like the FORM methods. A Monte Carlo simulation draws samples
directly from the probability distribution of the random variables and
generates the probability space of the output variables through integration.
One of these methods is Importance Sampling (IS)( Karamchandani et al
1989). The basic idea of importance sampling is to minimize the total number
of sampling points by concentrating on the sampling in the failure region
where the probability density is the greatest. However, in many cases, it is
difficult to know the shape of the failure region, in advance. To overcome this
difficulty, the concept of Adaptive Importance Sampling (AIS) has been
proposed (Bucher 1988, Melchers 1989).
The AIS is based on the idea that the importance-sampling density
function can be gradually refined to reflect the increasing state of knowledge
of the failure region. The sampling space is adaptively adjusted based on the
generated sampling points. Two versions of AIS have been developed. The
48
first version uses an adaptive surface to approximate the limit state. Based on
different adaptive surfaces, a radius-based method, a plane-based method, and
a curvature-based method have been developed (Wu 1998). The second
version of AIS is called multimodal adaptive importance sampling (Wu 1998,
Zou et al 2002). It uses a multimodal sampling density to emphasize all
important sample points in the failure domain, each in proportion to the true
probability density at the particular sampling point (Zou et al 2002). This
method is applied to the component and system reliability analyses of large
structures (Zou et al 2002, Mahadevan and Dey 1997), Mahadevan and
Raghothamachar 2000) with very satisfactory results. To address the high
computational cost of the Monte Carlo method, several more-efficient
simulation-based methods have been developed (Haldar and Mahadevan
2000, Qu and Haftka 2004), McDonald and Mahadevan 2008). Zou et al
(2002) proposed a Reliability-based design method using simulation
techniques and an efficient optimization approach.
2.7.3 Response Surface Methodology
Response Surface Methodology (RSM) is developed to reduce the
computational burden of RBDO, by replacing the original failure function
g(x) by an equivalent function R(X) by which computational procedure can
be simplified maintaining the accuracy (Ranganathan 1990). While
developing the new model, it is important that it allows an easy and efficient
computation of the failure function under the loading/system condition but
still preserves the essential features of the system. This new mathematical
model representing the original limit state function is called the response
surface (Breitung 1996). The representation of the limit state function by the
response surface should be independent of the properties of the basic
variables involved. However, for improving the efficiency and accuracy of the
method including the subsequent reliability analysis, some prior knowledge of
49
the stochastic properties of the variables is to be used. The limit state surface
can be represented in a polynomial form as given below:
2i
n
1iii
n
1ii XcXba)X(R
(2.6)
where Xi , i = 1,2,…..n are the basic variables and parameters a, bi, ci
i= 1,2.. n are the constants to be determined.
Methods such as orthogonal designs or small composite designs
have been considered to fit second-order probabilistic responses in RBDO. It
has been found that RSMs developed for deterministic design optimization
are not suitable for the reliability analysis or RBDO. Lancaster (1986)
proposed a new RSM that is specifically suited for a reliability analysis which
is based on the moving least square method and design experiments. The
moving least square method better approximates the implicit response by
imposing a variable weight over a compact support. In the literature most
RSMs have been developed utilizing only the response data, and little attempt
has been made to use both the response and sensitivity data to construct
approximate responses (Roux et al 1998, Zheng and Das 2000). An RSM that
utilizes only response data may be acceptable as far as the response values are
concerned; however, the approximate design sensitivity obtained from the
approximate response may contain sufficient error to cause difficulty in
RBDO. If accurate sensitivity information can be obtained efficiently, it can
be incorporated in the proposed moving least square method to substantially
improve the accuracy of the approximate responses, as well as the sensitivities
required for RBDO (Kharmanda et al 2002, Byeng and Choi 2004, Kaymaz
and McMohan 2004, Qu and Haftka 2004).
50
2.7.4 Hybrid Method
A hybrid reliability method combines the best features of
FORM/SORM, MCS and response surface approaches to achieve both
accuracy and efficiency (Zou et al (2002). Youn et al (2003) presented the
hybrid mean value method to adaptively select either the AMV method or the
CMV method, once the performance function type is identified. In order to
reduce the high computational time of the nested problems, Kharmanda et al
(2002), proposed a new formulation hybrid design space by combining
deterministic and random spaces. Mohsine et al (2006) have proposed a
modification of the formulation of the hybrid method to improve the optimal
solutions. The proposed method is called the Improved Hybrid Method
(IHM).
2.8 RBDO SOFTWARES
Numerous computer programs have been developed by researchers
to implement the FORM/SORM procedures. NESSUS (Numerical Evaluation
of Stochastic Structures Under Stress), developed at the Southwest Research
Institute combines probabilistic analysis with a general purpose finite
element/boundary element code. Design analysis is performed using the
displacement method, the mixed-iterative formulation or the boundary
element method, and the iterative perturbation is used for the sensitivity
analysis (Wu 1998). PROBAN (PROBability ANalysis) is developed at Det
Norske Veritas (Hovik, Norway). It is designed to be a general probabilistic
analysis tool. PROBAN is capable of estimating the probability of failure
using the FORM or SORM. The approximate FORM/SORM results can be
updated through importance sampling. The probability of general events can
be computed by the Monte Carlo simulation and directional sampling.
51
CALREL (CAL-RELiability) is a general-purpose reliability
analysis program designed to compute probability integrals. It incorporates
four general techniques for computing the probability of failure, namely,
FORM, SORM, directional simulation with exact or approximate surfaces,
and Monte Carlo simulation. Khalessi et al (1993) developed FEBREL (Finite
Element-Based RELiability) as a general-purpose, probabilistic, finite
element computer program at Rockwell International Corporation's Space
System Division. This software uses the ANSYS general purpose finite
element computer program to provide the necessary computational framework
for analyzing complex structures, while the FEBREL reliability computer
program provides the basis for modeling, analysis of uncertainties, and
computation of probabilities.
Frangopol and Estes (1998) developed RELSYS (RELiability of
SYStems) to compute the system reliability of structures modeled as a
series-parallel combination of its components. A probabilistic fracture
mechanics code called DARWIN (Darwin’s user guide 2006) has been
developed to predict the risk of fracture associated with rotors and disks
containing material anomalies.
2.9 RBDO APPLICATIONS
Several applications of Reliability Based Design Optimization are
reported in the literature.
2.9.1 Civil Structure
RBDO methods have been applied to a wide range of design and
maintenance problems in civil and architectural engineering (Frangopol 1997,
Enright and Frangopol 1998). The methods have been successfully
52
disseminated into several areas of civil applications such as buildings,
bridges, nuclear and off-shore structures (Davidson et al 1980, Feng and
Moses 1986). Enright and Frangopol (1999) studied the condition of
reinforced concrete girder bridges, using a time-variant system reliability
approach, in which both load and resistance are time-variant quantities.
Several system models are considered, including failure of any girder (series
system) and failure of a specified number of adjacent girders (series-parallel
system). Adaptive importance sampling is used to determine the
cumulative-time system failure probability. The influence of resistance
degradation and post-failure load redistribution is included. Pettit and Grandhi
(2000) addressed the use of a light weight composite modular deck to replace
the existing deteriorated reinforced concrete deck.
2.9.2 Automotive Industry
Yang et al (2002) applied the reliability-based optimal design to the
crashworthiness design of a full vehicle system in multi-crash scenarios. They
demonstrated that the weight could be reduced compared with a deterministic
(baseline) design, while satisfying the safety constraints. Zou et al (2002)
proposed an efficient method for the reliability analysis of a vehicle
body-door subsystem with respect to one of the important quality issues—the
door closing energy. The developed method combines the optimization-based
and simulation-based approaches and is particularly applicable for problems
with highly non-linear and implicit limit state functions. Byeng and Choi
(2004) proposed a new methodology for RBDO by integrating the RSM and
the HMV method of PMA and applied this method for a large-scale
application of the design of vehicle side impact.
53
2.9.3 Aerospace Industry
Uncertainty is introduced primarily on the conceptual design level,
where reliability analysis methods are combined with system level
deterministic analyses (Rao 1986, Yang et al 1990). Yang and Nikolaidis
(1991) applied RBDO to a preliminary design of the wing of a small
commuter airplane subjected to gust loads that are modeled using
probabilistic distributions. Grandhi and Wang (1998) applied optimization to
minimize the weight of a twisted gas turbine blade subjected to a probabilistic
constraint on natural frequency with the consideration of uncertainties in the
material properties and thickness distributions. A general framework for the
stochastic multi-disciplinary aircraft design is presented in accounting for
various sources of uncertainties, such as modeling and economic variability,
and aiming for system affordability (Mavris and DeLaurentis 1998, Leverant
et al 2003). Qu et al (2000) applied RBDO to a hydrogen storage tank under
cryogenic temperature. Stroud et al (2002) proposed the probabilistic design
of a plate-like wing to meet flutter and strength requirements.
2.9.4 Composite Materials
Composite materials are being widely used in modern structures,
such as aircraft and space vehicles, because of their high performance, high
temperature resistance, tailoring facility, and light weight. Considerable
research has been carried out on the design and failure analysis of composite
structures (Thanedar and Chamis 1995, Chao 1996). Several applications of
RBDO methods to the design of composite thin walled structures have been
reported in the literature (Richard and Perreux 2000, Biagi and Medico 2008).
Yang and Ma (1989), Miki et al (1993) and Su et al (2002) have considered
probabilistic load conditions and material properties including manufacturing
54
uncertainties. Antonio et al (1996) considered Composite structures with
degradation models and buckling instabilities.
Kogiso et al (1997) applied the RBDO to a symmetric laminated
plate. The reliability based optimization procedure is developed and applied
to minimize the weight of eight fiber reinforced polymer composite bridge
deck panel configurations (Enright and Frangopol 1999). Mahadevan and
Liu (1998) proposed a probabilistic optimum design for composite laminates.
Qu et al (2000) applied RBDO for composites under a cryogenic
environment. The results of experiments and research into composite
materials show large statistical variations in their mechanical properties (Lin
2000). Therefore, probabilistic analysis plays an important role in reliability
assessment (Nicholaidis et al 2008).
2.9.5 Other Applications
Pu et al (1997) applied RBDO to a typical frame of a
small-waterplane-area twin-hull ship subjected to system-reliability
constraints on failure criteria, considering uncertainties on the loads and
material strength. Yang et al (2005) solved an exhaust system problem using
different RBDO methods and the results are compared. Kharmanda and
Olhoff (2004) applied RBDO for Topology Optimization. Alan et al (2007)
focuses on improving the design and reliability of robotic systems by
addressing the uncertainties in the operational point.
The percentage contribution of the RBDO study in different
applications is shown in Figure 2.6.
55
Figure 2.6 RBDO in different Applications
2.10 DESIGN OPTIMIZATION OF COMPOSITE LAMINATES
The scope of composite materials in engineering design and the use
of composite material design in various real life scenarios, are extensively
reviewed (Agarwal and Broutman 1990). Cheng et al (1980) analyzed more
generally, the buckling problems of non-homogeneous anisotropic cylindrical
shells under combined axial, radial and torsional loads with all four boundary
conditions at each end of the cylinder. Methods are proposed by Nshanian and
Pappast (1983) for the determination of the optimal ply angle variation through
the thickness of symmetric angle-ply shells of uniform thickness. Weeton et al
(1986) briefly described the application possibilities of composites in the field
of the automotive industry to manufacture composite elliptic springs, drive
shafts and leaf springs. Beard and Johnson (1986) have discussed the potential
for composites in automotive applications. The Shell theory, based on the
critical speed analyses of drive shafts, has been presented by Dos Reis et al
(1987). Pollard (1989) studied the possibility of using polymer Matrix
composites in driveline applications. Patricia (1990) investigated the dynamic
behavior of supercritical composite drive shafts for helicopter applications.
Faust et al (1990) described the considerable interest on the part of both the
helicopter and automobile industries in the development of lightweight drive
shafts. Haftka and Walsh (1992) discussed extensively the stacking-sequence
56
optimization for buckling of laminated plates by integer programming.
Kim et.al (1992) minimized the weight of composite laminates with ply drop
under a strength constraint. Rajeev and Krishnamoorthy (1992) proposed a
method for converting a constrained optimization problem into an
unconstrained optimization problem.
Serge (1994) examined the optimum design of laminated plates and
shells subjected to constraints of strength, stiffness, buckling loads, and
fundamental natural frequencies. Ganapathi and Varadan (1994) studied
extensively the nonlinear free flexural vibrations of laminated circular
cylindrical shells. A method of analysis involving Kirchoff-Love’s first
approximation theory and Ritz’s procedure, and is used to study the influence
of boundary conditions and fiber orientation on the natural frequencies of thin
orthotropic laminated cylindrical shells, is presented (Lam and Toy, 1995).
A first order theory is presented by Lee (1995) to determine the
natural frequencies of an orthotropic shell. A theoretical analysis is presented
for determining the buckling torque of a cylindrical hollow shaft with layers of
arbitrarily laminated composite materials by means of various thin-shell
theories (Bert and Kim 1995). Lien-Wen Chen et al (1998) analyzed the
stability behaviour of rotating composite shafts under axial compressive loads.
Bauchau et al (1998) measured the torsional buckling loads of graphite/epoxy
shafts, which are in good agreement with theoretical predictions based on a
general shell theory including elastic coupling effects and transverse shearing
deformations.
Riche and Haftka (1993) studied the use of the Genetic Algorithm to
optimize the stacking sequence of composite laminates for buckling load
maximization. Various genetic parameters including the population size, the
probability of mutation, and the probability of crossover are optimized by
57
numerical experiments. Park et al (2001) used Genetic Algorithms for the
optimal design of symmetric composite laminates, subject to various loading
and boundary conditions. Kim et al 2001 studied an adhesively bonded joint
for composite propeller shaft. The main features of Genetic Algorithms and
the several ways in which they can solve difficult design problems, such as the
design of composite materials, are discussed by Gabor and Ekart (2003). The
General Motors pickup trucks, which adopted the composite drive shaft
(Spicer product), enjoyed a demand three times that of the projected sales in its
first year (Lee et al 2004).
2.11 CONCLUSION BASED ON REVIEW
The following observations are made from the literature review:
Uncertainties are inherently present in all real life engineering
systems. To ensure high reliability and safety, uncertainties
inherent to or encountered by the product during the entire life
must be considered in the design process (Oberkampf et al
2004). It is found from the literature that design optimization
methodologies should account for the stochastic nature of
engineering systems.
In traditional deterministic design optimization methods, the
uncertainty is handled through the safety factor, which does
not provide any scientific meaning and results in more
conservative designs (Qu and Haftka 2004). The literature
review clearly shows that there is a need to use new design
optimization methods such as the Robust Design Optimization
and Reliability Based Design Optimization (RBDO) that
incorporate uncertainty in the engineering design.
58
In general, it is found that the Robust Design optimization
method yields designs which are insensitive to uncertainties
(Frangopol 2003). The literature review reveals that
Reliability Based Design Optimization is a more rational
approach that quantifies the reliability or risk of failure in
probabilistic terms and includes these terms directly in design
optimization as reliability constraints (Zou and Mahadevan
2006).
Many researchers have used reliability estimation methods
like the First Order Reliability Method (FORM) and the
Second Order Reliability Method (SORM) in Reliability
Based Design Optimization study (Zhao and Ono 1999).
These approximation methods do not guarantee optimal
solutions for highly non-linear limit state functions. Monte
Carlo Simulation based methods are more suitable to solve
such problems (Mahadevan and Dey 1997).
Composite materials find wide applications in industries such
as Aerospace and Automotive industries seeking ways to
reduce the weight, because these materials have higher
strength to weight and stiffness to weight ratios. These
composite materials exhibit large variations in the Material
and Geometric properties (Philippids 2000). Several
researchers have reported the use of Reliability Based Design
Optimization study in Civil structure and Aerospace industry
applications (Papadrakakis 1998, Stroud et al 2002). But the
application of RBDO for composite material design is
scarcely examined.
Reliability Based Design Optimization applications to
determine the optimum stacking sequence of the composite
59
plates, cylindrical shell and pipes is reported in the literature
(Kogiso 1997, Enright and Frangopol 1999). The application
of RBDO for composite drive shaft design is seldom
investigated.
The composite drive shaft design optimization problem is
considered in this study. The design of composite laminates is a complex
combinatorial optimization problem. It is difficult to solve this problem using
traditional mathematical programming techniques. Hence an attempt is made
to solve the design optimization problem using search heuristics such as the
Genetic Algorithm, the Particle Swarm Optimization and the Evolutionary
Programming.
2.12 SUMMARY
Reliability Based Design Optimization considers the uncertainties
to quantify the risk and reliability of the components or the system at the
design level. In the literature review on traditional design optimization,
various types and models of handling design uncertainties and a review on
design methodologies under uncertainty are reported. An overview of the
formulation, methodologies and applications of RBDO is also presented. The
literature review reveals that there is an increase in interest among researchers
to explore the use of RBDO in real-life applications. Though a lot of research
has been carried out in the area of RBDO, there appears to be a lack of
attention in RBDO applications for composite materials. Composite materials
offer an excellent strength-to-weight ratio, and also find wide applications in
automotive and aerospace industries. The application of heuristics for the
reliability based design optimization of a composite drive shaft is carried out
in this study.